On the Range of Exponential Functionals of Lévy Processes

Abstract

We study the exponential functional ∞ 0 e −ξ s− dη s of two one-dimensional independent Lévy processes ξ and η, where η is a subordinator. In particular, we derive an integro-differential equation for the density of the exponential functional whenever it exists. Further, we consider the mapping Φ ξ for a fixed Lévy process ξ, which maps the law of η 1 to the law of the corresponding exponential functional ∞ 0 e −ξ s− dη s , and study the behaviour of the range of Φ ξ for varying characteristics of ξ. Moreover, we derive conditions for selfdecomposable distributions and generalized Gamma convolutions to be in the range. On the way we also obtain new characterizations of these classes of distributions.2010 Mathematics subject classification. 60G10, 60G51, 60E07.

On the law of killed exponential functionals

“…2.1]) that, provided ξ and η are not both deterministic, it has an invariant probability distribution if and only if the stochastic integral t 0 e −ξs− dη s converges almost surely to a finite limit as t → ∞ (see, e.g., [14] for necessary and sufficient conditions), in which case the limit random variable V 0,ξ,η := ∞ 0 e −ξs− dη s := lim t→∞ t 0 e −ξs− dη s is called the exponential functional of (ξ, η). Due to this connection, the law of V 0,ξ,η is well-studied in the literature see, e.g., [10], [12], the survey by Bertoin and Yor [9], or [4], [5], [8], [16], [21] for some more recent results.…”

“…Introducing jumps of size −1 to the process U by adding an atom with mass q > 0 to the Lévy measure of U , or equivalently considering U = U − N , where N denotes an independent Poisson process with parameter q > 0, Equation (1.2) yields a Lévy process ξ that is killed upon the first jump of N , i.e., after an exponential time. The stochastic differential equation dX t = X t− d U t + dη t , t ≥ 0, (1.3) also has a solution that is unique in law and a Markov process (see [3] and [7]). However, the stationary distribution of the process is now given by the killed exponential functional of (ξ, η) with parameter q V q,ξ,η := τ 0 e −ξs− dη s , (1.4) (cf.…”

“…In [3], different distributional equations were derived through methods such as Laplace inversion to study both the law and the density of the exponential functional in the case where q = 0 and η is a subordinator. In [21], similar equations were derived in the case where q > 0, η t = t and ξ is a subordinator, also establishing properties of the density such as the limiting behavior at t = 0.…”

“…From this, we calculate the infinitesimal generator of the process, which is the starting point of the analysis. An integro-differential equation for the characteristic function of the killed exponential functional is then derived in Section 4 using the methods from [3] and [4]. Section 5 is concerned with deriving a general equation for the law of V q,ξ,η through Schwartz theory of distributions using a similar approach to [16], also discussing applications and special cases such as the case without killing.…”

Exponential functionals of Markov additive processes

“…Due to this connection, the resulting importance in applications, and their complexity, exponential functionals have gained a lot of attention from various researchers over the last decades, see e.g. [5,6,8,20,22,23] to name just a few.…”